Robust Control of DC-DC Boost Converter by using µ-Synthesis Approach

2019 IFAC Workshop on 2019 IFAC Workshop Control of Smart Gridon and RenewableAvailable Energy Systems online at www.sciencedirect.com 2019 IFAC Workshop on Control of Smart and Renewable Energy Systems 2019 Workshop on Jeju, IFAC Korea, JuneGrid 10-12, 2019 Control of Smart Grid and Renewable Energy Systems Jeju, Korea, June 10-12, 2019 Control of Smart Grid and Renewable Energy Systems Jeju, Korea, June 10-12, 2019 Jeju, Korea, June 10-12, 2019

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Robust Control of DC-DC Boost Converter Robust Control of DC-DC Boost Converter Robust Control of DC-DC DC-DC Boost Converter using µ-Synthesis Approach Robustby of Boost Converter byControl using µ-Synthesis Approach by using µ-Synthesis Approach by using µ-Synthesis B. Alharbi* M. Alhomim*Approach R. McCann*

B. Alharbi* M. Alhomim* R. McCann*  B. R.  B. Alharbi* Alharbi* M. M. Alhomim* Alhomim* R. McCann* McCann* * University of Arkansas, Fayetteville, AR 72701 USA (e-mail: [email protected]) * University of Arkansas, Fayetteville, AR 72701 USA (e-mail: [email protected]) ** University University of of Arkansas, Arkansas, Fayetteville, Fayetteville, AR AR 72701 72701 USA USA (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Boost converters are used in many renewable energy systems in order to interface a lower Abstract: Boostsuch converters arearray used or in battery many renewable energyofsystems in order toinverter. interface a lower voltage source as a solar to the dc input a grid-connected The boost Abstract: Boost converters are used in many renewable energy in order to interface aa lower Abstract: Boostsuch converters are used or in many renewable energy systems in order toinverter. interface lower voltage source as a solar array battery toforthe dceffects input ofsystems a grid-connected The boost converter must be controlled in order to correct the of variable input voltages and changing voltage source such as a solar array or battery to the dc input of a grid-connected inverter. The boost voltage source such as a solar array or battery to the dc input a grid-connected inverter. The boost converter must be controlled in order to correct for the effects of variable input voltages and changing output loads. This paper develops an improved controller designofmethod using a voltages µ-synthesis approach. converter must be controlled in order to correct for the effects variable input and changing converter must be controlled in order to correct for the effects of variable input voltages and changing output loads. This paper develops an improved controller design method using a µ-synthesis approach. The goal is toThis achieve adevelops low order controller that can bedesign implemented at a reduced cost. This is an output loads. paper an improved controller method using aa µ-synthesis approach. output loads. paper an improved controller method using µ-synthesis approach. The goal is toThis achieve adevelops low order controller that can performance bedesign implemented at more a reduced cost.realizations. This is an controllers that achieve robust with complex improvement over H ∞ ∞ a low order controller that can be implemented at a reduced cost. This is an The goal is to achieve The goal is to achieve a low order controller that can be implemented at a reduced cost. This is an improvement over H controllers that achieve robust performance with more complex realizations. ∞ Detailed simulation results for a wide range of parametric changes in the boost circuit environment improvement over H ∞ controllers that achieve robust performance with more complex realizations. improvement over H controllers that achieve robust performance with more complex realizations. Detailed simulation results for a wide range of parametric changes in the boost circuit environment ∞ confirm the benefits of the proposed method. The results are compared with a PI controller whereby it is Detailed simulation results for aa wide range of parametric changes in the circuit environment Detailed simulation results for improved wide range of results parametric changes in thea boost boost circuit whereby environment confirmthat the benefits of the proposed method. The are compared with PI controller it is shown µ-synthesis provides performance compared to a conventional controller. confirm the benefits of the proposed method. The results are compared with a PI controller whereby confirm theµ-synthesis benefits of provides the proposed method. The resultscompared are compared with a PI controller whereby it it is is shown that improved performance to a conventional controller. shown that µ-synthesis provides improved performance compared to a conventional controller. © 2019,that IFAC (International Federation of Automatic Control) Hosting Elsevier Ltd. controller. All rights reserved. shown µ-synthesis provides improved performance compared to by aµ-synthesis. conventional Keywords: Boost converter, robust control, structured singular value, Keywords: Boost converter, robust control, structured singular value, µ-synthesis. Keywords: singular  Keywords: Boost Boost converter, converter, robust robust control, control, structured structured singular value, value, µ-synthesis. µ-synthesis. 

 1. INTRODUCTION 2. ROBUST CONTROLLER DESIGN  1. INTRODUCTION 2. ROBUST CONTROLLER DESIGN 1. 2. 1. INTRODUCTION INTRODUCTION 2. ROBUST ROBUST CONTROLLER CONTROLLER DESIGN DESIGN Robustness in control systems has been a focal point for Robustness in control systems has been a focal point for 2.1 Robust Design Method designers since the 1980s when George Zames and Bruce Robustness in systems been aa focal point for Design Method Robustness in control control systems has been Zames focaland point for 2.1 Robust designers since the 1980s whenhas George Bruce Robust Design Method Francis first introduced the importance of the and topicBruce and 2.1 designers since the 1980s when George Zames 2.1 Robust Method models provide advantages designers since the 1980sthe when George Zames Francis first introduced importance of the and topicBruce and State-space Design time-domain extended it quickly to more general problems. Currently, Francis first introduced the importance of the topic and State-space time-domain models provide advantages Francis introduced the importance of the Currently, topic and extendedfirst it quickly to more general problems. to time-domain frequency domain descriptions. In particular, for compared State-space models provide robust control design totends to general emphasize H∞∞ techniques extended it quickly more problems. Currently, State-space time-domain models provide In advantages advantages compared to frequency domain descriptions. particular, extended it quickly to more general problems. Currently, robust control design tends to emphasize H techniques for ∞ analysis and design of multiple-input multiple-output optimization ofdesign performance objectives. The process is based compared to frequency domain descriptions. In particular, robust control tends to emphasize H ∞ techniques for compared to frequency domain descriptions. In particular, analysis and design of multiple-input multiple-output robust controlof design tends single-input to emphasize H∞process techniques for (MIMO) optimization performance objectives. The is(SISO) based can often be handled in a more compact manner. on the theory that a suitable single-output analysis and design of multiple-input multiple-output optimization of performance objectives. The process is based analysis and design of multiple-input multiple-output (MIMO) can often be handled in a more compact manner. optimization of performance objectives. The process is(SISO) based on the theory that a suitable single-input single-output Designing a system that has robust performance to structured controller provides sufficient gain and phase margin. The (MIMO) often be in aa more manner. on the theory that aa suitable single-input single-output (SISO) (MIMO) can often that be handled handled in performance more compact compact manner. Designingcan a system has robust to structured on the theory that suitable single-input single-output (SISO) controller provides sufficient gain and phase margin. The and unstructured uncertainty if generally very difficult. There closed-loop achievessufficient robust stability byphase solving a linear Designing a system that has robust performance to structured controller provides gain and margin. The Designing a system thatresearch has robust performance to structured and unstructured uncertainty if generally very difficult. There controller provides sufficient gain and phase margin. The closed-loop achieves robust stability by solving a linear has been considerable on this topic. Lin’s defined matrix inequality constraint with a defined optimization cost and unstructured uncertainty if generally very difficult. There closed-loop achieves robust stability by solving aa linear and unstructured uncertainty if generally veryimportant difficult. There has been considerable research on this were topic. Lin’s defined closed-loop achieves robust stability by solving linear matrix inequality constraint with a defined optimization cost specifications and methods (Lin, 2000) in this function (Doyle, 1982). has been considerable research on this topic. Lin’s defined matrix inequality constraint with a defined optimization cost has been considerable research on this topic. Lin’s defined specifications and methods (Lin, 2000) were important in this matrix constraint with a defined optimization cost study: functioninequality (Doyle, 1982). H , Loop Transfer Recovery (LTR) method, µ∞ ∞ specifications and methods (Lin, 2000) were important in this function (Doyle, 1982). specifications and methods (Lin, 2000) were important in this study: H , Loop Transfer Recovery (LTR) method, µ∞ function Robust (Doyle, control 1982). has been developed over the previous synthesis, and Quantitative Feedback Theory (QFT) methods. study: H ∞, Loop Transfer Recovery (LTR) method, µRobust control has been developed over the previous study: H , Loop Transfer Recovery (LTR) method, µsynthesis, and Quantitative Feedback Theory (QFT) methods. ∞ decades to guarantee stability and high performance of synthesis, and Quantitative Feedback Theory (QFT) methods. Robust control has been developed over the previous Robust control has been developed over the previous decades to guarantee stability and high performance of synthesis, and Quantitative Feedback Theory (QFT) methods. electric power converters with and respect toperformance uncertainties, decades to stability high of decades to guarantee guarantee stability highsuch of 2.2 Perturbation Models electric power converters with and respect toperformance uncertainties, as magnetic 2.2 Perturbation Models disturbance effects, and nonlinearities electric power converters with respect to uncertainties, electric power converters with respect toshaping, uncertainties, disturbance effects, andsensitivity, nonlinearities such as magnetic saturation. Using mixed H∞∞ loop and µ- 2.2 2.2 Perturbation Perturbation Models Models disturbance effects, and nonlinearities such as magnetic disturbance effects, and nonlinearities such as magnetic saturation. Using mixed sensitivity, H∞ Postlethwaite, loop shaping, and µ- Although nominal models are generally not accurate for synthesis approaches (Skogestad and 2005), Although nominal not accurate for saturation. Using mixed sensitivity, H ∞ loop shaping, and µoperation, theymodels can be are usedgenerally to effectively illustrate the saturation. Using mixed sensitivity, H∞ Postlethwaite, loop shaping, µ- actual synthesis approaches (Skogestad and 2005), Although nominal models are generally not accurate for one can model uncertainty to find satisfactory solutions.and Although nominal models are generally not accurate for actual operation, they can be used to effectively illustrate the synthesis approaches (Skogestad and Postlethwaite, 2005), modelsoperation, of the plant wherein input voltage and load are related synthesis approaches (Skogestad and Postlethwaite, one can model uncertainty to find satisfactory solutions.2005), actual they can be used to effectively illustrate the actual operation, they can bezero usedofvoltage tothe effectively illustrate the models of the plant wherein input and load are related one can model uncertainty to find satisfactory solutions. to dc gain, poles, and RHP system. At this point, one can model uncertainty satisfactory solutions. of the plant wherein input and related The µ-synthesis approach to is find a model for robust control and models ofand the plantand wherein input voltage and load load are related to dc gain, poles, RHP zero ofvoltage the system. At are thisSimple point, The µ-synthesis approach is a model for robust control and models inductor capacitor values will have little impact. to dc gain, poles, and RHP zero of the system. At this point, one of the successful techniques for designing an efficient The µ-synthesis approach is a model for robust control and to dc gain, poles, and RHP zero of the system. At this point, inductor and capacitor values will have little impact. Simple The µ-synthesis approach is a model robust an control and full one of the successful techniques for for designing efficient block perturbation will will include all the effects of inductor and capacitor values have little impact. Simple controller within structured and unstructured uncertainty and one of the successful techniques for designing an efficient inductor and capacitor values will have little impact. Simple full block perturbation will include all the effects of one of thewithin successful techniques for designing an efficient controller structured and unstructured uncertainty and electromagnetic properties at low and high frequency. To full block perturbation will include all the effects of disturbance conditions. Thisand method was proposed by Doyle controller within structured unstructured uncertainty and full block perturbation will include all the effects of electromagnetic properties at low and high frequency. To controller structured unstructured uncertainty and achieve disturbancewithin conditions. Thisand method was proposed by Doyle this, an input multiplicative perturbation is appliedTo to electromagnetic properties at low and high frequency. in 1982, and helps toThis assess the was impact of parametric disturbance conditions. method proposed by Doyle electromagnetic properties at low and high frequency. To achieve this, an input multiplicative perturbation is applied to disturbance conditions. method proposed by design Doyle aachieve in 1982, and helps toThis assess the was impact of parametric nominal model: As shown in (1) where ∆(s) is a unity norm uncertainty. In this research, µ-synthesis controller this, an input multiplicative perturbation is applied to in 1982, and helps to assess the impact of parametric achieve this, an input multiplicative perturbation is applied to aperturbation nominal model: As shown in (1) where ∆(s) is a unity norm in 1982, and helps to assess the impact of parametric uncertainty. In this research, µ-synthesis controller design and As w(s) is in an(1)uncertainty weight whose methods are used as aresearch, structuredµ-synthesis singular value approach. aaperturbation nominal model: shown where ∆(s) is a unity norm uncertainty. In this controller design nominal model: As shown in (1) where ∆(s) is a unity norm and w(s) is an uncertainty weight whose uncertainty. In this controller design perturbation methods are used as aresearch, structuredµ-synthesis singular value approach. magnitude displays the isuncertainty bound weight (Baskinwhose and and an methods are used singular value approach. and w(s) w(s) an uncertainty uncertainty magnitude displays the isuncertainty bound weight (Baskinwhose and methods are develops used as as aa structured structured singular value approach. This paper an improved robust controller design perturbation Bulent, 2015), magnitude displays the uncertainty bound (Baskin This paper develops an improved robust controller design magnitude displays the uncertainty bound (Baskin and and Bulent, 2015), method using µ-synthesis to achieve optimal voltage This paper develops an robust design 2015), This paper develops an improved improved robust controller controller design Bulent, method using µ-synthesis to achieve optimal voltage Bulent, 2015), G ( s)  G( s)(1  ( s) w( s)) . regulation of a boost converter used in renewable energy (1) method µ-synthesis to achieve optimal method using µ-synthesis to used achieve optimal voltage voltage regulationusing of a boost converter in renewable energy G ( s)  G( s)(1  ( s) w( s)) . (1) applications. Boost converters are particularly challenging regulation of aa boost converter used in energy G ss))  G (( ss)(1   (( ss)) w (( ss)) .. (1) ( regulation of Boost boost converterare used in renewable renewable energy The nominal model applications. converters particularly challenging G (  G )(1   w )) (1)1  and possible plant set is shown in Table due to a tendency toconverters become unstable at high boost ratios. applications. Boost are particularly challenging applications. Boosttoconverters are particularly challenging due to a tendency become unstable at high boost ratios. The nominal model and possible plant set is shown in Table and the nominal setand corresponding these plants is in shown in1 Using an averagingtotechnique, the evaluation of the method due to aa tendency become unstable at high ratios. The nominal model possible plant set is shown Table due to an tendency totechnique, become unstable at functions highofboost boost ratios. Using averaging the evaluation thethat method The nominal model and possible plant set is shown in Table 1 and the nominal set corresponding these plants is shown in1 considers power conversion and weighting will Fig. the1. nominal The uncertainty boundthese andplants possible input Using an averaging technique, the evaluation of the method and set corresponding is shown in Using an averaging technique, the evaluation of the method considers power conversion and weighting functions that will and set corresponding is shown in Fig. the1. nominal The uncertainty boundthese andplants possible input improve robustness, stability, and performance. considers power functions considers power conversion conversion and weighting functions that that will will Fig. improve robustness, stability,and andweighting performance. Fig. 1. 1. The The uncertainty uncertainty bound bound and and possible possible input input improve robustness, stability, and performance. 2405-8963 © 2019, IFACstability, (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. improve robustness, and performance.

Copyright 2019 responsibility IFAC 223Control. Peer review©under of International Federation of Automatic Copyright © 2019 IFAC 223 10.1016/j.ifacol.2019.08.261 Copyright © 2019 IFAC 223 Copyright © 2019 IFAC 223

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multiplicative perturbations is shown in Fig. 2 where the uncertainty bound covers all possible values.

201

transfer functions; loops can be shaped by weighting functions. Closed-loop systems require stability and performance and H∞ has the ability to assure that. The H∞ method requires that the system transfer function being evaluated as the norm (Tewari, 2002)

2.3 H-infinity Method A stable robust system meets specific criteria even in the existence of defined uncertainties. The H∞ optimization method, first defined in the 1980s, has proven itself to be exceedingly efficient as a successful robust control design method in the field of linear time-invariant control systems as Ioannou and Sun explained (Ioannou and Jing, 1996). H∞ methods enable robust stable performance based on normbased optimization theory and are intended to perform well even in worst-case disturbance events.

G



 sup w{ G( jw) }

(2)

This functions to decrease the peak of the bode diagram that then raises the margin of robust stability. This study uses Matlab software to minimize the norm from the weighted mixed-sensitivity functions. 2.4 Mixed Sensitivity Problem For robust stability a controller will be defined by a mixed sensitivity problem

Table 1. Boost converter parameters Parameters Values Units Input Voltage, Vi 20 ± 20% Voltage, V Output Voltage, Vo 45 Voltage, V Inductor, L 560  10-6 Henry, H Capacitor, C 100  10-6 Farad, F Resistance load, R 10 ± 20% Ohm, Ω

S min K KS



Wp ( I  GK )1  min K KWn ( I  GK )1

(3) 

which describes the closed-loop system with model uncertainties and possible perturbations shown in Fig. 3 (Pilat and Wlodarezyk, 2011). Plant, P y1 W1

y2 W2

u u1

y3

G

W3

-

K

Fig. 3. Mixed sensitivity control design. The system G refers to a transfer function of the boost converter. P is the open-loop transfer function. W1, W2 and W3 are the performance, control and noise weighting functions respectively. The input signals are denoted by u, y is the output vector, and K is the controller that can be found by using MATLAB commands

Fig. 1. Effect of the parameters variation on gain and phase.

[K,CLP,gamma,info]=mixsyn(G,W1,W2,W3) This penalizes the error signal W1, control signal,W2, and output signal,W3, so that the closed-loop transfer function matrix is weighted mixed sensitivity as

Ty1u1

W1S   W2 R  W3T 

(4)

where the sensitivity, S, is equal to (I+GK)-1, the complementary sensitivity, T, is equal to (GKS), and the control effort, R, is equal to (KS). The three functions of sensitivity, control effort, and complementary sensitivity will be shaped using mixed sensitivity method.

Fig. 2. Bound of uncertainty and input multiplicative perturbations. Design specifications, including tracking and robustness performances, used constraints on the singular values of loop 224

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 ( N (s))  sup  ( N ( jw)), w  R

The control purpose is to design a stable controller that minimizes the norm of generalized transfer function Tyw such that

Tyw



 1 or

W1S W2T

1

(8)

1

where   ( N ) is the lowest singular value of ∆. This indicates a frequency dependent stability margin.

(5)



2.5 µ-synthesis Consider the general plant and controller configuration presented in Fig. 4. Here P(s) is the generalized plant model that contains all of known elements encompassing the nominal plant model and performance and uncertainty weighting functions. K(s) is a stabilizing controller, and ∆(s) is the norm-bounded block-diagonal perturbation matrix, w represents an external disturbance, v is the measurement available to the controller, u is the output from the controller, and ∆y is an error signal.

N Fig. 5. N∆ structure for robust performance analysis. Next, to perform robust stability analysis, matrix C(s) is defined such that C(s) = N11(s)

u

y

u

y

(transfer function from the output to the input ∆(s)).

P

v

u

y

u K

Fig. 4. General control configuration with uncertainty

C

(s)  {diag i ( s) : i ( s)  C, i  1, 2,..n} . (6) Fig. 6. C∆ structure for robust stability analysis Since C(s) is stable, µ is defined as,

Assume that ∆(s) is stable and

(s) s  jw is normalized such that ( jw)



 1.

 (C( jw))  [min{ (i ( jw)) : det( I  C( jw)( jw)  0}]1

(9)

The output of the general control configuration are an observation output vector v, and a performance scale vector z, that is desired to be small. The inputs of the configuration are a control input u, and a system disturbance vector w that includes all inputs external to the system such as scaled reference signals, disturbance, and sensor noise signals. ∆y and ∆u are the input and output signals of the dynamic uncertainties, respectively.

∆(s). To maintain robust stability, two conditions should be met:

For a given controller K(s), the system is consolidated as shown in Fig. 5, where

For robust performance, the conditions are (i) robust stability, and (ii) µ(N(jw))<1, w .

Here µ (C(jw)) is the structured singular value of

C(s) s  jw , measuring the smallest structured uncertainty

N (s) s  jw is stable and (ii) µ(C(jw))<1, w .

The goal of µ-synthesis design methodology is to find the stabilizing controller K that minimizes the influence of the external disturbances w, on the performance output z, in the case of a perturbation ∆ and that satisfies

N (s)  ( P( s), K (s)) P11 (s)  P12 ( s) K ( s)( I  P22 ( s) K ( s)) 1 P21 ( s) . (7) Definition 1: When N(s) is an interconnected transfer matrix as in Fig. 5, the structured singular value with respect to ∆ is defined by:

Fl [ Fu ( P,), K 225



 1.

(10)

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203

3. CONTROLLER DESIGN FOR BOOST CONVERTER 3.1 Controller Concept The controller K can be designed by solving Riccati equations (Skogestad et al., 2005), or by a series of LMIs (Dullerud and Fernardo, 2013), which mimics the standard µsynthesis procedures in robust control theory. After the design process is done entirely, the robust controller is fulfilled using µ-synthesis method by solving the mixed sensitivity problem. 3.2 Weighting Functions Selection Fig. 7. Weighting function performance. According to (5), the significance of the magnitude response of S lies beneath the magnitude response of 0.8, and the magnitude response of T should lie less than the response of 0.75, Fig. 8 reveals that these conditions are proved using H∞ controller. The objective of the weighting functions is to manifest the importance of fulfilling performance criteria within significance frequency range.

Designing the appropriate weights for the plant in order to achieve a desired attenuation at certain frequencies is the challenging task in robust controller design. If

G( jw) K ( jw) is large enough at frequencies of

expected disturbance, the sensitivity of the output of those frequencies will be very small and the disturbance will be rejected. The weight factor is needed to have high enough gains at disturbance frequencies in order to achieve the desired loop shape. The general guidelines for weighting functions choice used in this paper and proposed in (Skogestad and Postlethwaite, 2005) and (Gu et al., 2005) W1 

s  wb Mp

3.3 Robust Analysis The robustness properties can be evaluated by implementing the appropriate µ-tests for the uncertain feedback system displayed in Fig. 9. The transfer functions Wdel and ∆ parameterize the multiplicative uncertainty at the converter input. Presume the transfer function Wdel is known, and the transfer function ∆ is stable.

(11)

s  wb  e p

where W1 is the bound for the sensitivity function and reflects external disturbance rejection, a steady state error. ep is the maximum steady-state offset, wb is the demanded bandwidth and Mp is the peak value of the sensitivity, therefore W1 

0.7 s  6300 s 1

W2 scales KS that mirror the control input which affects the converter slightly. To avoid this impulsive input effect on the converter, W2 is chosen as W2=0.001. The complementary sensitivity function T can be shaped by W3 which must be large at high frequencies to limit T for a stable system, s W3 

wb Mp

.

Fig.8. Mixed sensitivity control design results.

(12)

e p s  wb

To shorten the closed-loop bandwidth, Mp is selected small and to maintain the stability of the system, the complementary sensitivity, needs to be low at high frequency range. For that, ep is selected as 0.1µ and thus

Delta*Wdel

y

r Controller

+

W3 

s  6200 . s  120

+

System

-

Fig. 7 shows the frequency response of weighting functions W1 and W3.

Fig. 9. Closed-loop system with input multiplicative uncertainty. 226

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The uncertainty weight Wdel is chosen as

Wdel 

s  7 107 s  7 108

3.4 Robust Stability and Performance Theorem 1: (Robust stability with µ) Suppose

N  Fl ( P, K ) is robustly stable with reference to

 i with i



 1 , if and only if supw i ( N ( jw))  1 .

Robust stability of the plant was checked based on theorem 1. The µ-value for robust stability is (0.4005) which means the allowable structured perturbations with norm less than (1/0.4005). MATLAB code robstab reports that the closedloop system remains stable up to 4 times of the possible values.

Fig. 11. Performance analysis with worst case conditions.

4. MATLAB/SIMULATION RESULTS Applying ‘dksyn’ from MATLAB robust control toolbox and parameters stated in Table 1, the controller is given by

Theorem 2: (Robust performance with µ) Suppose

N  Fu ( N , i ) with stable lower fractional

A K ( s)   k Ck

transaction (N) that satisfies the performance condition

Fu ( N , i )   1 for all  i with i



 1, if and only if

supw  ( Fl ( P, K )( jw))  1 .

Bk  Dk 

(13)

where

1 0 0 0     0 64 0 120  Ak    2.463 109 3.008 106 3.008 107 0.01202   0 0 4444 1000  

The robust performance is achieved based on Theorem 2, if and only if, the closed-loop frequency response is less than one for each frequency calculated. The robust performance test for the controller is shown in Fig. 10, which shows robust performance is achievable with peak µ value (0.4005) which shows that the allowable structured uncertainties with norm less than 1/0.3002.

0.1508  0   Bk    0     0 

These finding correspond with the worst case procedure as shown in Fig. 11, which ensures the stability of the plant in existence of parameter variations.

Ck  5.854 108 7.149 105 2.183 106 188.6 Dk  0 Four iterations were performed resulting peak µ-value and gamma. Table 2 shows the summery of iteration. Table 2. DK-iteration summery Iteration # Controller Order Total D-Sclae Order Gamma Achieved Peak mu-Value

1 7 0 669.6634 0.778

2 51 44 24.908 0.654

3 17 10 4.200 0.726

4 21 14 35.695 11.415

The robust stability and robust performance characteristics of the closed-loop system can be evaluated. According to the results, the controller achieves robust performance with µ= 0.4005. The step response of µ-synthesis controller is shown

Fig. 10. Assessment of robust performance.

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in Fig. 12. The µ-synthesis controller shows better dynamic responses over PID controller as shown in Fig. 13. The model is simulated by using SIMULINK (Fig.14) that compares both controllers. The output voltages is displayed in Fig. 15. It is obviously shown that µ-synthesis controller has less overshoot and less time settling. The load is changed at 0.04 and 0.08 seconds while the system still stable.

Fig. 15. Simulation result for output voltage. 5. CONCLUSION A robust controller design method using µ-synthesis for uncertain models of a boost converter circuit was derived. It exhibits several improvements compared to traditional design methods. In particular, a fast step-response to input voltage disturbances and rapid settling times were achieved. The effectiveness of the design method were demonstrated through detailed simulation results using Simulink-SimPower Systems. It was found that the structured singular value, µ, is useful for controlling boost converter circuits. REFERENCES Fig. 12. Step response of the controller.

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Fig. 13. Comparison of PID and µ-synthesis controllers.

Fig. 14. Boost closed loop schematic simulation with Matlab/Simulink.

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